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Implicit difference approximation for the time fractional diffusion equation. (English) Zbl 1140.65094
Author’s summary: We consider a time fractional diffusion equation on a finite domain. The equation is obtained from the standard diffusion equation by replacing the first-order time derivative by a fractional derivative (of order \(0<\alpha<1\)). We propose a computationally effective implicit difference approximation to solve the time fractional diffusion equation. Stability and convergence of the method are discussed. We prove that the implicit difference approximation (IDA) is unconditionally stable, and the IDA is convergent with \(O(\tau+h^2)\), where \(\tau\) and \(h\) are time and space steps, respectively. Some numerical examples are presented to show the application of the present technique.

MSC:
65R20 Numerical methods for integral equations
45J05 Integro-ordinary differential equations
26A33 Fractional derivatives and integrals
35K05 Heat equation
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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